PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Introduction Timetable Assignments...

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1 PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Introduction Timetable Assignments...5

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3 i PMTH212, Multivariable Calculus Assignment Summary 2009 For External Students Assignment Date to be Posted Description Number 1 3 March Problems 2 10 March Problems 3 17 March Problems 4 24 March Problems 5 31 March Problems 6 7 April Problems 7 8 May Problems 8 15 May Problems 9 22 May Problems May Problems 11 5 June Problems June Problems This assignment summary may be detached for your notice board.

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5 1 PURE MATHEMATICS 212 Multivariable Calculus Introduction Textbook and Lecture Notes The textbook is Calculus by H. Anton, 6th or 7th edition, John Wiley & Sons. The lecture notes you have received cover all the material required of this unit. It is essential that you read carefully these lecture notes and try to understand everything presented there. While the booklet of lecture notes is more concise, the above mentioned textbook contains more material and also many exercise questions. Therefore the lecture notes and the textbook can be used to complement each other. The following textbooks are recommended as reference books, where you can find slightly different ways of presenting the topics covered by this unit and also some more material related to this unit. Multivariable Calculus by J. Stewart, 4th edition, Brooks/Cole Publishing Company. Multivariable Calculus, concepts and contexts by J. Stewart, Brooks/Cole Publishing Company. Calculus of Several Variables by R.A. Adams, 2nd edition, Addison-Wesley Publishers Ltd. Assessment There will be a three-hour examination in the June-July examination period. The final assessment consists of two parts: your assignment solutions will be counted as 20% and the June-July examination will be counted as 80%. Lectures Three lectures per week during the First Semester. Past experience shows that those internal students who do not come to most of the lectures usually fail the unit at the end. Therefore, if you are an internal student, I strongly urge you to attend the lectures. Tutorials One tutorial per week mainly to help internal students with their assignments. Assignments Assignments are an essential part of this unit. It is important to notice that through doing the assignments and through the feedbacks from your marked assignments you

6 2 can check your understanding of the material in the unit in a timely fashion. As poor understanding of earlier material generally makes the study of the later part of the unit much more difficult, I urge you try not to get too far behind, and not to leave important material unfinished before going to the next step. It is important that you submit your assignments at the scheduled times. Late assignments, especially towards the end of the semester, may not be marked. If you are stuck with a particular question in your assignment, try to get help. I am happy to be able to help you with any of your questions. If after considerable efforts, you still cannot do it, then it is better to submit your assignment without answering that particular question than having the assignment rather late. You will receive complete solutions to the assignment problems when the marked assignments are returned. If you are external, your assignments should go through the Teaching and Learning Center so as to be recorded as received by the University. Unit Outline This unit is a logical extension of the calculus studied in the first-year mathematics units. It mainly consists of Chapters of Anton, 7th edition (Chapters in the 6th edition). The material for Lecture 17, Parametric Problems, is not covered by Anton. The specific topics covered by this unit are listed in the timetable on pages 3-4 below. You might find it helpful to have your first year calculus notes handy, as the first year material is frequently used in this unit. In particular, if you find yourself rather rusty with the first year differentiation and integration techniques, it might be a good idea to have a review of them before you engage in working on the partial derivatives and multiple integrals in this unit. Instructions for Examination The examination questions will be concerned with basic concepts and general methods. The style will be similar to the examples in the lecture notes, and the questions in the assignments and the past examination papers. Past exam papers with solutions will be sent to you near the end of the semester. In the exam, you are allowed to bring with you three A4 sized pieces of paper of handwritten notes. Both sides of the paper can be used. Unit Coordinator If you have any specific questions concerning the work, or the unit in general, please don t hesitate to contact me. You can reach me by phone on: (02) or syan@turing.une.edu.au. You may fax if necessary on (02) My office is B163. You are most welcome to come to see me on matters related to this unit. Wish you enjoyment and success in your studies. Dr. Shusen Yan

7 3 TIMETABLE Submission dates for external assignments are given in the Assignment Summary page. For internal students, Wednesday is the due day for the assignment based on the material taught in the previous week. The section numbers referred to below are based on the 7th edition of Anton, followed by the numbers of the corresponding material in the 6th edition in brackets. Lecture Topic Sections in Anton 1 Coordinates and Graphs in 3-space 12.1 (13.1, 13.2) 2 Vectors, Dot Product, Cross Product (13.3, 13.4) 3 Lines and Planes 12.5, 12.6 (13.5, 13.6) Assignment 1 4 Quadric Surfaces 12.7 (13.7) 5 Vector Valued Functions 13.1, 13.2 (14.1, 14.2) 6 Integration of Vector Valued Functions and Arc Length 13.2, 13.3 (14.3) Assignment 2 7 Unit Tangent and Normal Vectors 13.4 (14.4) 8 Curvature 13.5 (14.5) 9 Multivariable Functions 14.1 (15.1) Assignment 3 10 Limits and Continuity 14.2 (15.2) 11 Partial Derivatives 14.3 (15.3) Assignment 4 12 Differentiability of Two Variable Functions 14.4 (15.4) 13 The Chain Rules 14.5 (15.4) 14 Tangent Planes and Total Differentials 14.7 (15.5) Assignment 5 15 Directional Derivatives and Gradients 14.6 (15.6) 16 Functions of Three and n Variables 14.1, 14.3 (15.7) 17 Parametric Problems Not in Anton

8 4 Assignment 6 18 Maxima and Minima 14.8 (15.8) 19 Extrema Over a Given Region 14.8 (15.8) 20 Lagrange Multipliers 14.9 (15.9) Assignment 7 21 Double Integrals 15.1 (16.1) 22 Iteration of Double Integrals 15.2 (16.2) 23 Double Integrals in Polar Coordinates, Surface Area 15.3,15.4 (16.3, 16.4) Assignment 8 24 Triple Integrals 15.5 (16.5) 25 Change of Variables 15.8 (16.8) 26 Triple Integrals in Cylindrical and Spherical Coordinates 15.7, 15.8 (13.8, 17.4) Assignment 9 27 Line Integrals 16.2 (17.2) 28 Line Integrals Independent of Path 16.3 (17.3) 29 Green s Theorem 16.4 (17.4) Assignment Surface Integrals 16.5 (17.5) 31 Surface Integrals of Vector Functions 16.6 (17.6) 32 The Divergence Theorem 16.7 (17.7) Assignment Stokes Theorem 16.8 (17.8) 34 Applications 16.8 (17.8) 35 Revision Lecture Assignment 12

9 5 ASSIGNMENT 1 1. Show that the points (2, 1, 6), (4, 7, 9) and (8, 5, 6) are the vertices of a right triangle. 2. Find the distance from the point (5, 2, 3) to the (a) xy-plane (b) xz-plane (c) y-axis. 3. Find the terminal point of v=i+2j 3k if the initial point is ( 2, 1, 4). 4. Find a vector n which is perpendicular to the plane determined by the points A(0, 2, 1), B(1, 1, 2), and C( 1, 1, 0). 5. Find parametric equations for the line in R 2 through (1,1) and parallel to the line x = 5 + t, y = 1 2t; 6. Determine whether the planes are perpendicular. (a) x y + 3z 2 = 0, 2x + z = 1 (b) 3x 2y + z = 1, 4x + 5y 2z = Find equations of the planes. (a) Through ( 1, 4, 3) and perpendicular to the line x 2 = t, y + 3 = 2t, z = t (b) Through ( 1, 2, 5) and perpendicular to the planes 2x y + z = 1 and x + y 2z = 3.

10 6 ASSIGNMENT 2 1. Name and sketch the surface: z 2 = 4x 2 + y 2 + 8x 2y + 4z 2. Find an equation of the orthogonal projection onto the xy-plane of the curve of intersection of the paraboloid z = 4 x 2 y 2 and the parabolic cylinder z = y 2 3. Describe the graph of the vector equation. (a) r=(3 sin2t)i + (3 cos 2t)j. (b) r = 2i + tj + (t 2 1)k. 4. Determine whether is a smooth curve of the parameter t. r = cos(t 2 )i + sin(t 2 )j + e t k 5. Let u,v,w be differentiable vector-valued functions of t. Prove that d du dw [u [v w)] = [v w] + u [dv w] + u [v dt dt dt dt ] 6. (a) Evaluate [(t sin t)i + j] dt; (b) Find the arc length of the curve r(t) = (3 cost)i + (3 sin t)j + tk; 0 t 2π.

11 7 ASSIGNMENT 3 1. Find the unit tangent vector T and the unit normal vector N for the given value of t: x = e t cost, y = e t sin t, z = e t ; t = Find the curvature at the indicated point: x = e t, y = e t, z = t; t = Show that for a plane curve described by y = f(x) the curvature κ(x) is κ(x) = d 2 y/dx 2 [1 + (dy/dx) 2 ] 3/2 [Hint: Let x be the parameter so that r(x) = xi + yj = xi + f(x)j.] 4. Let f(x, y) = x + (xy) 1/3. Find (a) f(t, t 2 ), (b) f(x, x 2 ), (c) f(2y 2, 4y). 5. Find g(u(x, y), v(x, y)) if g(x, y) = y sin(x 2 y), u(x, y) = x 2 y 3, v(x, y) = πxy. 6. Let f(x, y, z) = zxy + x. Find (a) f(x + y, x y, x 2 ), (b) f(xy, y/x, xz). 7. Describe the level surfaces. (a) f(x, y, z) = 3x y + 2z (b) f(x, y, z) = z x 2 y 2.

12 8 ASSIGNMENT 4 1. Find the region where the function f is continuous. (a) f(x, y) = (x y) 1/2. xy (b) f(x, y) = cos( 1 + x 2 + y 2). 2. Find the limit, if it exists. (a) (b) x 4 16y 4 lim (x,y) (0,0) x 2 + 4y. 2 sin(x 2 + y 2 ) lim. (x,y) (0,0) x 2 + y 2 x 3 y 3. (a) Show that the value of approaches 0 as (x, y) (0, 0) along any straight 2x 6 + y2 line y = mx, or along any parabola y = kx 2. x 3 y (b) Show that lim does not exist. (x,y) (0,0) 2x 6 + y2 [Hint: Let (x, y) (0, 0) along the curve y = x 3, and then compare the result with the results in (a).] 4. Find f x, f y and f z : f(x, y, z) = z ln(x 2 y cosz). 5. (a) Show that f(x, y) = ln(x 2 + y 2 ) satisfies 2 f/ x f/ y 2 = 0. This is called the Laplace s Equation. (b) Show that u(x, y) = e x cosy and v(x, y) = e x sin y satisfy u/ x = v/ y, u/ y = v/ x. These are called the Cauchy-Riemann Equations. 6. Let f(x, y) = (x 2 + y 2 ) 2/3. Show that f x (0, 0) = 0.

13 9 ASSIGNMENT 5 1 Let f(x, y) = e xy2. Find f xyx, f xxy and f yxx and verify their equality. 2. Let z = 3x 2y, where x = u + v ln u, y = u 2 v ln v. Find z/ v and z/ u by the chain rule. 3. Show that if u(x, y) and v(x, y) satisfy the Cauchy-Riemann equations (See Assignment 4, Question 5), and if x = r cosθ and y = r sin θ, then u r = 1 v r θ, v r = 1 u r θ. 4. Find equations for the tangent plane and normal line to the given surfaces at the point P: z = ln[(x 2 + y 2 ) 1/2 ]; P( 1, 0, 0). 5. Find a unit vector u that is perpendicular at P(2, 3) to the level curve of f(x, y) = 3x 2 y xy through P.

14 10 ASSIGNMENT 6 1. Find a unit vector in the direction in which f(x, y, z) = (x 3y +4z) 1/2 increases most rapidly at P(0, 3, 0), and find the rate of increase of f in that direction. 2. Let f be a function of one variable and let z = f(x 2 + y 2 ). Show that y z/ x x z/ y = Assume that F(x, y, z) = 0 defines z implicitly as a function of x and y. Show that z/ x = F/ x, z/ y = F/ y F/ z F/ z. 4. Let F(u, v) = v u f(t) dt. Show that F/ u = f(u), F/ v = f(v);

15 11 ASSIGNMENT 7 1. Locate all relative maxima, relative minima and saddle points: f(x, y) = x 2 + xy 2y 2x Find the absolute extreme of the given function on the indicated set R: f(x, y) = xy 2x; R is the triangular region with vertices (0,0), (0,4) and (4,0). 3. Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint: f(x, y) = x 2 y; x 2 + y 2 = Find the point on the line 2x 4y = 3 that is closest to the origin.

16 12 1. Evaluate the iterated integral: ASSIGNMENT 8 1 x 0 0 e x2 dy dx. 2. Evaluate the double integral. (a) (x + y) da; R is the region enclosed between the curves y = x 2 and y = x 1/2. (b) R R x cosy da; R is the triangular region bounded by y=x, y=0 and x=π. 3. Evaluate the integral by first reversing the order of integration. (a) (b) y/2 3 lnx 1 0 cos(x 2 ) dxdy, xdy dx. 4. (a) Use polar coordinates to evaluate the first quadrant within the circle x 2 + y 2 = 9. R (9 x 2 y 2 ) 1/2 da, where R is the region in (b) Evaluate the iterated integral by converting to polar coordinates. 2 2 (4 y 2 ) 1/2 (4 y 2 ) 1/2 e (x2 +y2) dxdy. 5. Find the surface area of the portion of the plane 2x + 2y + z = 8 in the first octant that is cut off by the three coordinate planes.

17 13 1. Evaluate the triple integral ASSIGNMENT 9 R cos(z/y) dv, where R is defined by π/6 y π/2, y x π/2, 0 z xy. 2. Express the integral as an equivalent integral in which the z-integration is performed first, the y-integration second and the x-integration last: 3 (9 z 2 ) 1/2 (9 y 2 z 2 ) 1/ f(x, y, z) dxdy dz. 3. Let G be the rectangular box defined by a x b, c y d, k z l. Show that G b d l f(x)g(y)h(z) dv = [ f(x) dx][ g(y) dy][ h(z) dz]. a c k 4. (a) Use cylindrical coordinates to find the volume of the solid bounded above and below by the sphere x 2 + y 2 + z 2 = 9 and laterally by the cylinder x 2 + y 2 = 4. (b) Use spherical coordinates to find the volume of the solid bounded above by the sphere ρ = 4 and below by the cone φ = π/3.

18 14 ASSIGNMENT Let F=(3x+2y)i+(2x-y)j. Evaluate C F dr where C is (a) the line segment from (0,0) to (1,1); (b) the curve y = x 2 from (0,0) to (1,1) 2. Find the work done by a force F(x,y,z)=xyi+yzj+xzk acting on a particle that moves along the curve r(t) = ti + t 2 j + t 3 k, 0 t Show that integral by (π, 1) (0,1) y sin xdx cosxdy is independent of the path, and evaluate the (a) using the fundamental theorem of line integrals (Theorem in Anton); (b) integrating along the line segment from (0,1) to (π, 1). 4. Show that the integral (1,π/2) (0,0) find its value by any method. 5. Use Green s Theorem to calculate e x sin y dx + e x cosy dy is independent of the path, and C (e x + y 2 )dx + (e y + x 2 ) dy, where C is the boundary of the region between y = x 2 and y = x oriented counterclockwise.

19 15 ASSIGNMENT Evaluate the surface integral σ xyz ds where σ is the portion of the plane x+y+z=1 lying in the first octant. 2. Evaluate the surface integral σ (x2 + y 2 ) ds, where σ is the portion of the sphere x 2 + y 2 + z 2 = 4 above the plane z = Evaluate σ F n ds where F(x, y, z) = (x+y)i+(y+z)j+(z+x)k and σ is the portion of the plane x+y+z = 1 in the first octant, oriented by unit normals with positive components. 4. Use the Divergence Theorem to evaluate σ F nds, where n is the outer unit normal to σ. (a) F(x, y, z) = 2xi + 2yj + 2zk, σ is the sphere x 2 + y 2 + z 2 = 9. (b) F(x, y, z) = z 3 i x 3 j + y 3 k, σ is the sphere x 2 + y 2 + z 2 = a 2.

20 16 ASSIGNMENT Use Stokes Theorem to evaluate C F dr, where F(x, y, z) = 3y 2 i + 4zj + 6xk, C is the triangle in the plane z = (1/2)y with vertices (2, 0, 0), (0, 2, 1) and (0, 0, 0), with a counterclockwise orientation looking down the positive z-axis. 2. Show that if the components of F and their first- and second-order partial derivatives are continuous, then div(curlf) = Find the net volume of the fluid that crosses the surface σ in the direction of the orientation in one unit of time where the flow field is given by F(x, y, z) = x 2 i+yxj+zxk, σ is the portion of the plane 6x+3y+2z=6 in the first octant oriented by upward unit normals.